Classical Geometries in Modern Contexts Geometry of Real Inner Produ
The focus of this book and its geometric notions is on real vector spaces X that are finite or infinite inner product spaces of arbitrary dimension greater than or equal to 2. It characterizes both euclidean and hyperbolic geometry with respect to natural
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Classical Geometries in Modern Contexts Geometry of Real Inner Product Spaces Third Edition
Walter Benz FB Mathematik Mathematisches Seminar Universität Hamburg Hamburg Germany
ISBN 978-3-0348-0419-6 ISBN 978-3-0348-0420-2 (eBook) DOI 10.1007/978-3-0348-0420-2 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2012945178 © Springer Basel 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper
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Contents Preface
ix
Preface to the Second Edition
xiii
Preface to the Third Edition
xv
1
2
Translation Groups 1.1 Real inner product spaces . . . . . . . . . . 1.2 Examples . . . . . . . . . . . . . . . . . . . 1.3 Isomorphic, non-isomorphic spaces . . . . . 1.4 Inequality of Cauchy–Schwarz . . . . . . . . 1.5 Orthogonal mappings . . . . . . . . . . . . 1.6 A characterization of orthogonal mappings . 1.7 Translation groups, axis, kernel . . . . . . . 1.8 Separable translation groups . . . . . . . . . 1.9 Geometry of a group of permutations . . . . 1.10 Euclidean, hyperbolic geometry . . . . . . . 1.11 A common characterization . . . . . . . . . 1.12 Other directions, a counterexample . . . . .
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